---
title: 'Fast Fixed-Effects Estimation: Short introduction'
author: "Laurent Berge and Grant McDermott"
date: "`r Sys.Date()`"
output:
  html_document:
    theme: journal
    highlight: haddock
    number_sections: yes
    toc: yes
    toc_depth: 3
    toc_float:
      collapsed: yes
      smooth_scroll: no
  pdf_document:
    toc: yes
    toc_depth: 3
vignette: > 
  %\VignetteIndexEntry{fixest introduction} 
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

set.seed(0)

is_DT = requireNamespace("data.table", quietly = TRUE)

if(is_DT) library(data.table)

library(fixest)
setFixest_nthreads(1)
```

<!-- Help on vignette writing here: http://r-pkgs.had.co.nz/vignettes.html. -->

The package `fixest` provides a family of functions to perform estimations with multiple fixed-effects.
The two main functions are `feols` for linear models and `feglm` for generalized linear models. In addition, the function `femlm` performs direct maximum likelihood estimation, and `feNmlm` extends the latter to allow the inclusion of non-linear in parameters right-hand-sides. Finally, the functions `fepois` and `fenegbin` are aliases for Poisson and negative binomial fixed-effect estimations.
Each of these functions supports any number of fixed-effects and is implemented with full fledged multi-threading in c++. Functions `feols` and `feglm` further support variables with varying slopes.

This package is currently (Feb. 2020) the fastest software available to perform fixed-effects estimations. See the project's [homepage](https://lrberge.github.io/fixest/) for a set of benchmarks.

The standard-errors of the estimates can be easily and intuitively clustered (up to four-way).

The function `etable` allows to seamlessly export the results of multiple estimations into either a data.frame, or into a Latex table.

The main features of the package are illustrated in this vignette. The theory used to obtain the fixed-effects is based on [Berge (2018)](https://wwwen.uni.lu/content/download/110162/1299525/file/2018_13), "*Efficient estimation of maximum likelihood models with multiple fixed-effects: the R package FENmlm.*" CREA Discussion Papers, 13.

# Simple example using trade data

This example deals with international trade, which is a setup that usually requires performing estimations with many fixed-effects. We estimate a very simple gravity model in which we are interested in finding out the negative effect of geographic distance on trade. The sample data consists of European trade extracted from Eurostat. Let's load the data contained in the package:

```{r echo=TRUE}
library(fixest)
data(trade)

```

This data is a sample of bilateral importations between EU15 countries from 2007 and 2016. The data is further broken down according to 20 product categories. Here is a sample of the data:

```{r, echo=FALSE, results='asis'}
tab = head(trade)
knitr::kable(tab)
```


The dependent variable of the estimation will be the level of trade between two countries while the independent variable is the geographic distance between the two countries. To obtain the elasticity of geographic distance net of the effects of the four fixed-effects, we estimate the following:

$E\left(Trade_{i,j,p,t}\right)=\gamma_{i}^{Exporter}\times\gamma_{j}^{Importer}\times\gamma_{p}^{Product}\times\gamma_{t}^{Year}\times Distance_{ij}^{\beta}$,

where the subscripts $i$, $j$, $p$ and $t$ stand respectively for the exporting country, the importing country, the type of product and the year, and the $\gamma_{v}^{c}$ are fixed-effects for these groups. Here $\beta$ is the elasticity of interest. 

Note that when you use the Poisson/Negative Binomial families, this relationship is in fact linear because the right hand side is exponentialized to avoid negative values for the Poisson parameter. This leads to the equivalent relation:^[Since the $\gamma$ are parameters, I omit to put them in logarithmic form.]

$E\left(Trade_{i,j,p,t}\right)=\exp\left(\gamma_{i}^{Exporter}+\gamma_{j}^{Importer}+\gamma_{p}^{Product}+\gamma_{t}^{Year}+\beta\times \ln Distance_{ij}\right)$.


## Estimation

The estimation of this model using a Poisson likelihood is as follows:

```{r}
gravity_pois = fepois(Euros ~ log(dist_km) | Origin + Destination + Product + Year, trade)
```

The function `fepois` is actually an alias to the function `feglm` with `family = poisson`. The results can be shown directly with the `print` method:
```{r}
print(gravity_pois)
```

The `print` reports the coefficient estimates and standard-errors as well as some other information. Among the quality of fit information, the squared-correlation corresponds to the correlation between the dependent variable and the expected predictor; it reflects somehow the idea of R-square in OLS estimations.
Note that the estimation is performed using parallel computing which you can control using the argument `nthreads` (see the "multi-threading" section for more details).

## Clustering the standard-errors

To cluster the standard-errors, we can simply use the argument `vcov` of the `summary` method. Let's say we want to cluster the standard-errors according to the first two fixed-effects (i.e. the *Origin* and *Destination* variables). Then we just have to do:
```{r}
summary(gravity_pois, vcov = "twoway")
```
The clustering can be done on one, two, three or up to four variables. If the estimation includes fixed-effects, then by default the clustering will be done using these fixed-effects, in the original order. This is why the *Origin* and *Destination* variables were used for the two-way clustering in the previous example. If, instead, you wanted to perform one-way clustering on the *Product* variable, you need to provide it in a formula or use the argument `cluster`: 

```{r, eval = FALSE}
# Three ways to summon clustering on the Product variable
summary(gravity_pois, vcov = ~Product)
summary(gravity_pois, cluster = "Product")
summary(gravity_pois, cluster = ~Product)
```

Both produce the same results:
```{r, eval = TRUE}
summary(gravity_pois, cluster = ~Product)
```

Note that you can always cluster the standard-errors, even when the estimation contained no fixed-effect:

```{r}
gravity_simple = fepois(Euros ~ log(dist_km), trade)
# We use a formula to specify the variables used for two way clustering
# (note that the values of the variables are fetched directly in the original database)
summary(gravity_simple, ~Origin + Destination)
```

Finally, the standard-errors can also be computed at estimation time, you simply need to add the `vcov` argument:

```{r}
fepois(Euros ~ log(dist_km), trade, vcov = ~Product)
```


## Other standard-errors

Talking about standard-errors... there are more than clustered standard-errors that can be computed... and there are many ways to achieve the same thing... and many shortcuts to know. So before you leave don't forget to have a look at the section describing how to use the [vcov argument](#the-vcov-argument)!


## Other estimation functions

Now we estimate the same relationship by OLS. We need to put the left hand side in logarithm (since the right-hand-side is not exponentialized):

```{r}
gravity_ols = feols(log(Euros) ~ log(dist_km) | Origin + Destination + Product + Year, trade)
```

Of course you can use different families in `feglm`, exactly as in `glm`. 

To get the estimation for the fixed-effects Negative Binomial:

```{r}
gravity_negbin = fenegbin(Euros ~ log(dist_km) | Origin + Destination + Product + Year, trade)

```


## Viewing the results in R

Now let's say that we want a compact overview of the results of several estimations. The best way is to use the function `etable`. This function summarizes the results of several `fixest` estimations into a data.frame. To see the fixed-effects results with the three different likelihoods, we just have to type:

```{r, eval=FALSE}
etable(gravity_pois, gravity_negbin, gravity_ols,
         vcov = "twoway", headers = c("Poisson", "Negative Binomial", "Gaussian"))
```
```{r, echo=FALSE, results='asis'}
tab = etable(gravity_pois, gravity_negbin, gravity_ols, vcov = "twoway", headers = c("Poisson", "Negative Binomial", "Gaussian"))
# problem to display the second empty line in markdown
knitr::kable(tab[-2, ])
```

We added the argument `vcov="twoway"` to cluster the standard-errors for all estimations. As can be seen this function gives an overview of the estimates and standard-errors, as well as some quality of fit measures. The argument `headers` is used to add information on each estimation column.

In the previous example, we directly added the estimation results as arguments of the function `etable`. But the function also accepts lists of estimations.
Let's give an example. Say you want to see the influence of the introduction of fixed-effects on the estimate of the elasticity of distance. You can do it with the following code where we use the argument `fixef` to include fixed-effects (instead of inserting them directly in the formula):

```{r}
gravity_subfe = list()
all_FEs = c("Year", "Destination", "Origin")
for(i in 0:3){
	gravity_subfe[[i+1]] = fepois(Euros ~ log(dist_km), trade, fixef = all_FEs[0:i])
}
```

The previous code performs 4 estimations with an increasing number of fixed-effects and store their results into the list named `gravity_subfe`. To show the results of all 4 estimations, it's easy:

```{r, eval=FALSE}
etable(gravity_subfe, cluster = ~Origin+Destination)
```
```{r, echo=FALSE, results='asis'}
tab = etable(gravity_subfe, cluster = ~Origin+Destination)
knitr::kable(tab)
```

We have a view of the 4 estimations, all reporting two-way clustered standard-errors thanks to the use of the argument `cluster`.

## Multiple estimations

Note that since version 0.8.0, multiple estimations can be performed at once without requiring loops. Let's replicate the previous example using `fixest` stepwise functions:

```{r}
res_multi = fepois(Euros ~ log(dist_km) | csw0(Year, Destination, Origin), trade)
```

The previous line of code performs 4 estimations. The function `csw0` is the key here, it means: *cumulative stepwise starting with the empty element*. Starting with the empty element, each new estimation adds a new element in the `csw0()` function, quite like the previous loop. Then you can consider the results, here `res_multi`, as a list of results, although with specific methods to easily access each element. 

Stepwise functions can be applied to the linear right-hand-side and to the fixed-effects, you can also have multiple dependent variables and perform split sample estimations with the argument `split`. All of this is detailed in the dedicated vignette: [Multiple estimations](https://lrberge.github.io/fixest/articles/multiple_estimations.html).


## Exporting the results to Latex

So far we have seen how to report the results of multiple estimations on the R console. Now, using the same function `etable`, we can also export the results to high quality Latex tables. We just need to provide the argument `tex = TRUE`:

```{r}
# with two-way clustered SEs
etable(res_multi, cluster = ~Origin+Destination, tex = TRUE)
```

The user can export the Latex table directly into a file (argument `file`), add a title (arg. `title`) and a label to the table (arg. `label`).  Note that when the argument `file` is present, the Latex format becomes the default (i.e. `tex = TRUE` by default).

The coefficients can be renamed easily (arg. `dict`), some can be dropped (arg. `drop`) and they can be easily reordered with regular expressions (arg. `order`). 

The significance codes can easily be changed (arg. `signifCode`) and all quality of fit information can be customized (argument `fitstat`). Among others, the number of fixed-effect per fixed-effect dimension can also be displayed using the argument `fixef_sizes`.

### An elaborate example

Consider the following example of the exportation of two tables:

```{r, eval=FALSE}
# we set the dictionary once and for all
myDict = c("log(dist_km)" = "$\\ln (Distance)$", "(Intercept)" = "Constant")
# 1st export: we change the signif code and drop the intercept
etable(res_multi, signifCode = c("a" = 0.01, "b" = 0.05),
       drop = "Const", dict = myDict, file = "Estimation Tables.tex", 
       replace = TRUE, title = "First export -- normal Standard-errors")
# 2nd export: clustered S-E + distance as the first coefficient
etable(res_multi, cluster = ~Product, order = "Dist", 
       dict = myDict, file = "Estimation Tables.tex", 
       title = "Second export -- clustered standard-errors (on Product variable)")

```

In this example, two tables containing the results of the 4 estimations are directly exported to a Latex table into the file "Estimation Tables.tex". First take notice (again) that *we do not need to use the argument* `tex=TRUE` since when the argument `file` is present, the Latex format becomes the default. The file is re-created in the first exportation thanks to the argument `replace = TRUE`. 

To change the variable names in the Latex table, we use the argument `dict`. The variable `myDict` is the dictionary we use to rename the variables, it is simply a named vector. The original name of the variables correspond to the names of `myDict` while the new names of the variables are the values of this vector. Any variable that matches the names of `myDict` will be replaced by its value. Thus we do not care of the order of appearance of the variables in the estimation results. 

In the first export, the coefficient of the intercept is dropped by using `drop = "Const"` (could be anything such that `grepl(drop[1], "Constant")` is TRUE). In the second, the coefficient of the distance is put before the intercept (which is kept) thanks to the argument `order`. Note that the actions performed by the arguments `drop` or `order` are performed **after** the renaming takes place with the argument `dict`.

Note that you can completely customize the style of the table by using the `style` and `postprocessing` arguments, please have a look at the dedicated vignette: [Exporting estimation tables](https://lrberge.github.io/fixest/articles/exporting_tables.html). 


## Extracting the fixed-effects coefficients

To obtain the fixed-effects of the estimation, the function `fixef` must be performed on the results. This function returns a list containing the fixed-effects coefficients for each dimension. The `summary` method helps to have a quick overview:

```{r}
fixedEffects = fixef(gravity_pois)
summary(fixedEffects)
```

We can see that the fixed-effects are balanced across the dimensions. Indeed, apart from the first dimension, only one coefficient per fixed-effect needs to be set as reference (i.e. fixed to 0) to avoid collinearity across the different fixed-effects dimensions. This ensures that the fixed-effects coefficients can be compared within a given fixed-effect dimension. Had there be strictly more than one reference per fixed-effect dimension, their interpretation would have not been possible at all. If this was the case though, a warning message would have been prompted. Note that the mean values are meaningless per se, but give a reference points to which compare the fixed-effects within a dimension.
Let's look specifically at the `Year` fixed-effects: 

```{r}
fixedEffects$Year
```

Finally, the `plot` method helps to distinguish the most notable fixed-effects:

```{r, fig.width=7}
plot(fixedEffects)
```

For each dimension, the fixed-effects are first centered, then sorted, and finally the most notable (i.e. highest and lowest) are reported. The exponential of the coefficient is reported in the right hand side to simplify the interpretation for models with log-link (as the Poisson model). As we can see from the country of destination fixed-effects, trade involving France (FR), Italy (IT) and Germany (DE) as destination countries is more than 2.7 times higher than the EU15 average. Further, the highest heterogeneity come from the product category, where trade in product 4 (dairy products) is roughly 2.7 times the average while product 14 (vegetable plaiting materials) represents a negligible fraction of the average. 

Note however that the interpretation of the fixed-effects must be taken with extra care. In particular, here the fixed-effects can be interpreted only because they are perfectly balanced. 

# The vcov argument 

The version 0.10.0 of `fixest` introduced the argument `vcov`. This argument is highly versatile and single-handedly manages how the standard-errors are computed (note that you can find many example in [summary.fixest](https://lrberge.github.io/fixest/reference/summary.fixest.html#examples) help pages). This argument can be many things, so bear with me, and now, let's start trying to describe it!

## Basics: main calls and shortcuts

So far (version 0.10.0), six types of standard-errors can be computed, here are their keywords and a brief description:

 - `iid`: assumes that errors are identically and independently distributed.
 
 - `hetero`: assumes that errors are heteroskedastic (White correction).
 
 - `cluster`, `twoway`: clustered SEs, assumes that errors are correlated within the cluster groups.
 
 - `NW`, `newey_west`: Newey and West (1987) SEs for time series or panel data, assumes heteroskedastic and serially correlated errors.
 
 - `DK`, `driscoll_kraay`: Driscoll and Kraay (1998) SEs for panel data, assumes cross-sectionally and serially correlated errors.
 
 - `conley`: Conley (1999) SEs for cross-sections, assumes spatially correlated errors.
 
The most basic uses of the `vcov` argument are:
 
  - with a character scalar: `vcov = "vcov_type"` (ex: `vcov = "hetero"`).
  
  - with a formula of the form: `vcov = vcov_type ~ variables` where the variables are used to compute the SEs (ex: `vcov = DK ~ period`).
  
Most of the VCOV types need the information on some variables to be computed. For these types, one then needs to use a formula to inform on these variables. Let's give a first example:

```{r}
data(base_did)
est = feols(y ~ x1, base_did)
# Note that there is partial matching enabled (newey = newey_west)
summary(est, newey ~ id + period)
```

Here we queried Newey-West SEs and provided the panel identifiers in the right of the formula. Had we used the keyword only, that would have led to an error:

```{r, error=TRUE}
summary(est, "newey_west")
```

But, huh what does the error message says? That the `time` variable can be deduced? Indeed, that's the thing: the algorithm tries hard to provide sensible default values for the user.
Let's now try with panel identifiers set at estimation time:
```{r}
est_panel = feols(y ~ x1, base_did, panel.id = ~id + period)
summary(est_panel, "newey_west")
```
Oh, now it works! However I can see some shade of disappointment in your eyes: since we need to add `panel.id = ~id + period` we end up with more typing! Well, I'm not done: typically we tend to make many estimations with the same data set, so it may be useful to set some characteristics globally. This can be done with the `setFixest_estimation()` function:
```{r}
setFixest_estimation(panel.id = ~id + period)
est_implicit = feols(y ~ x1, base_did)
summary(est_implicit, "newey_west")
```
Although the exact same two lines of code led to an error a few paragraphs ago, now it works fine. 

Here is another example of implicit deduction:
```{r}
summary(est_implicit, "cluster")
```
The SEs are clustered at the unit identifier of the panel. Note that even if fixed-effects are present, the panel identifier takes precedence for clustering:
```{r}
feols(y ~ x1 | period, base_did, "cluster")
```
Of course, if the panel identifier is missing, the automatic clustering falls back to the first fixed-effect present:
```{r}
# Removing the panel
setFixest_estimation(reset = TRUE)
feols(y ~ x1 | period, base_did, "cluster")
```

Talking about clustered SEs, note that this is such a common operation that the empty keyword is equivalent to clustered SEs, so that `vcov = ~id + period` would lead to SEs clustered by `id` and `period`:
```{r}
feols(y ~ x1 | period, base_did, ~id + period)
```

Finally, a last example using Conley SEs:
```{r}
data(quakes)
feols(depth ~ mag, quakes, "conley")
```
In this case the latitude and longitude are deduced from the variables names in the data set used for the estimation (here these are `"lat"` and `"long"`). Note that, for convenience, an automatic distance cutoff is deduced from the data but is not likely to be the most appropriate! But how to pass your own cutoff as a parameter? That's what we'll see in the next section!


## Using dedicated functions

Some VCOVs require parameters: for example in the Newey-West VCOV, you need to provide the number of lags to consider. Although it is automatically deduced via a rule of thumb, the user is likely to want to use (or test) other values. 
To provide these extra parameters, you have two solutions:

 - use helper functions which have the same name as the VCOV keywords (ex: `NW` or `newey_west` are two functions). 
 
 - use dedicated VCOV functions (ex: `vcov_NW`).
 
We now cover these two methods in turn. Helper functions can be seen as supercharged keywords: they are used in place of the VCOV keywords but also accept arguments. Here are two examples:
```{r}
feols(y ~ x1 | period, base_did, NW(2) ~ id + period)

feols(depth ~ mag, quakes, conley(200, distance = "spherical"))
```
In the first example, we use the helper function `NW()` directly in the formula, exactly where the VCOV type would be. The first (and only) argument of `NW()` is the number of lags. On the right hand side of the formula, we give the variables used to identify the unit and the time variables, as usual.

In the second example, we use no formula but directly the helper function `conley()` for which we provide two arguments: the cutoff (first argument) and how the distance should be computed. The latitude and longitude are still deduced from the data. If it could not be deduced, we would have to provide a formula giving them, like in the previous example.

There are also dedicated VCOV functions which work in a more standard way. Let's redo the same two examples with them:
```{r}
feols(y ~ x1 | period, base_did, vcov_NW("id", "period", lag = 2))

feols(depth ~ mag, quakes, vcov_conley(lat = "lat", lon = "long", 
                                       cutoff = 200, distance = "spherical"))
```

Contrary to the helper functions: a) these functions cannot be used in a formula, b) they accept the variables to compute the SEs as arguments. They provide an alternative way to specify the VCOV.


## Small sample correction

The value of the SEs can (sometimes quite surprisingly) be impacted by the type of small sample correction (SSC) applied. In `fixest` you can monitor the SSC with the function `ssc()`. Details on how the SSCs are computed can be found in the dedicated vignette: [On standard-errors](https://lrberge.github.io/fixest/articles/standard_errors.html) -- here we detail only the implementation. 

Most `fixest` functions accept an `ssc` argument that you can readily use. Here's an example:
```{r}
est = feols(y ~ x1 | id, base_did)
est_up = feols(y ~ x1 | id, base_did, ssc = ssc(fixef.K = "full"))
est_down = feols(y ~ x1 | id, base_did, ssc = ssc(adj = FALSE, cluster.adj = FALSE))
etable(est, est_up, est_down)
```

Instead of providing the argument `ssc`, you can instead add the `ssc` call directly within the formula of the `vcov`. In this case you should simply add it as a variable.
We now replicate the previous example using a list of several VCOVs in `etable` (that's the only function that accepts lists in the argument `vcov`):
```{r}
etable(est, vcov = list(~id, ~id + ssc(fixef.K = "full"), 
                        ~id + ssc(adj = FALSE, cluster.adj = FALSE)))
```

You can also tweak `iid` or `hetero` SEs using a formula:
```{r}
feols(y ~ x1 | id, base_did, iid ~ ssc(adj = FALSE))
feols(y ~ x1 | id, base_did, hetero ~ ssc(adj = FALSE))
```


## Other types of VCOVs

Finally, we're approaching the end of the tour of the `vcov` argument. That argument can also accept:

 - a matrix, which will be used as the VCOV (possibly useful in specific contexts)
 
 - a function to be applied to a `fixest` object to compute the VCOV.

The matrix case needs no explanation, let's have a few lines on the function. You can provide custom functions to compute the VCOV:
```{r}
summary(est, vcov = sandwich::vcovHC, type = "HC1")
```
The argument `type` here is not an argument of `summary` but is passed directly to the function `vcovHC`. Note that in estimations you can still pass functions but they must not accept extra arguments. Hence creating you own wrapper function may be necessary:
```{r}
feols(y ~ x1 | id, base_did, vcov = function(x) sandwich::vcovHC(x, type = "HC1"))
```


# Instrumental variables

Two stage least square estimations can be performed by simply adding in the formula the link between the endogenous and the instrumental variables. Let's have an example.

```{r}
base = iris
names(base) = c("y", "x1", "x_endo_1", "x_inst_1", "fe")
set.seed(2)
base$x_inst_2 = 0.2 * base$y + 0.2 * base$x_endo_1 + rnorm(150, sd = 0.5)
base$x_endo_2 = 0.2 * base$y - 0.2 * base$x_inst_1 + rnorm(150, sd = 0.5)

est_iv = feols(y ~ x1 | x_endo_1 + x_endo_2 ~ x_inst_1 + x_inst_2, base)
est_iv
```

So we've just performed a two stage least squares estimation. The formula coming after the pipe, `x_endo_1 + x_endo_2 ~ x_inst_1 + x_inst_2`, describes the endogenous variables (on the left) and the instruments (on the right). 

By default, three statistics are displayed: the F-test from the first stage (weak instrument test), the Wu-Hausman endogeneity test and the overidentifying restrictions (Sargan) test. Note that the Sargan statistic appears only when relevant (i.e. when # instr. > # endo. vars., not the case here).

You can use the `fitstat` command to summon other kind of tests, notably Wald tests on the first/second stages:

```{r}
fitstat(est_iv, ~ ivf1 + ivwald1 + ivf2 + ivwald2, cluster = "fe")
```

As the Wald test relies on a given variance-covariance matrix, you can pass extra arguments to `fitstat`, as the argument `cluster` in the previous example, to specify which type of VCOV matrix is desired.

Note that you can display the statistics that you wish when printing by changing the default print values:

```{r}
setFixest_print(fitstat = ~ . + ivwald2)
est_iv
```

In the previous code, `fitstat = ~ . + ivwald2` means that we want to add the second stage Wald test to the existing printed statistics (represented here by the point). 

Now what about adding some fixed-effects? That's of course possible, you need to add them after the first right-hand-side, as follows:

```{r}
est_iv_fe = feols(y ~ x1 | fe | x_endo_1 + x_endo_2 ~ x_inst_1 + x_inst_2, base)
est_iv_fe
```

To access the first stage(s), you can use the summary method:

```{r}
summary(est_iv_fe, stage = 1)
```

When `summary` shall return more than one element, the object returned is not a regular `fixest` object but a `fixest_multi` object. These kind of objects are covered in the vignette: [Multiple estimations](https://lrberge.github.io/fixest/articles/multiple_estimations.html). You can display the first and second stages in a table with `etable`:

```{r}
etable(summary(est_iv_fe, stage = 1:2), fitstat = ~ . + ivfall + ivwaldall.p)
```

Here we use `stage = 1:2` to get all first stage regressions followed by the second stage. Using `stage = 2:1` would have done the opposite. 

Now some explanations regarding `fitstat`. The suffix `all` concerns IV only and means the following: if it's a first stage regression, then the first-stage F-stat is displayed, otherwise it's the second stage F-stat. The suffix `.p` is used in `ivwaldall.p` to access the p-value and not the statistic. Finally, you can permanently set which fit statistic to display in `etable` by using `setFixest_etable`, like for example `setFixest_etable(fitstat = ~ . + ivfall + ivwaldall.p)`.

# Interaction terms

Most R users will be familiar with the base expansion operators for creating model interaction terms, e.g. `x1*x2`, `x1:x2`, and `x1/x2`. These base operators all work with `fixest` models. However, the package also provides its own specialized syntax for creating interaction terms and combining variables. Relative to the base methods, these `fixest` methods offer significant performance gains and synergies with the package's other functions. To balance performance and convenience, the exact syntax depends on whether the interaction involves fixed-effects or not. Here we walk through both categories, further providing examples of common use-cases.

## Interactions involving fixed-effects

There are two reasons why we would want to interact variables in the fixed-effects slot. First, we may simply wish to combine fixed-effects (e.g. firm × country effects). Second, we wish to allow for varying slopes (e.g. a time trend for each firm). Let us consider each in turn, using a lightly modified version of the `iris` dataset:

```{r}
# Our base data for this section
base = iris
names(base) = c("y", paste0("x", 1:3), "fe1")
# Create another "fixed-effect"
base$fe2 = rep(letters[1:5], 30)
head(base)
```

### Combining several fixed-effects (`fe1^fe2^fe3...`)

Say we want to "combine" the two fixed-effect variables `fe1` and `fe2` to create a brand new fixed-effect variable. We can do it simply via `fixest`'s special `^` operator:

```{r}
est_comb = feols(y ~ x1 | fe1^fe2, base)
est_comb
```

The `^` operator does the same thing as `paste0(species, "_", fe)`, but is more convenient and significantly faster for large datasets. You can still extract the fixed-effects the same way:

```{r}
fixef(est_comb)[[1]]
```

Note further that more than two fixed-effects can be combined in exactly the same manner (e.g. `fe1^fe2^fe3`), and the syntax carries over to multivariate clustering too (e.g. `cluster = ~fe1^fe2^fe3`).

### Varying slopes (`fe[x]`)

You can introduce variables with varying slopes directly into the fixed-effects part of the formula using square brackets (`[]`). Recall that varying slopes allow us to flexibly control for heterogeneous effects across groups. Common real-life examples could be the inclusion of time trends for each observational unit (e.g. country), or allowing for the effect of some control variable (e.g. income) to be moderated by a fixed-effect (e.g. gender). Here we demonstrate by continuing with our simple dataset.

```{r}
head(base)
```

Say we want to estimate `y` as a function of `x1`, but controlling for `x2`. Moreover, we think that the slope coefficient of our `x2` control variable should be allowed to vary by the `fe1` fixed-effect variable. We can do this as follows:

```{r}
est_vs = feols(y ~ x1 | fe1[x2], base)
est_vs
```

If you want to see the slopes for `x2`, just use the function `fixef`:

```{r}
summary(fixef(est_vs))
```

Note that this produces the same result as `feols(y ~ x1 + fe1/x2, base)`, but is much more performant for large datasets. The trade-off from the `fe1[x2]` method is that we lose the ability to estimate confidence intervals on the resulting slope coefficients. This should be fine in cases where we are just trying to control for some confounding or nuisance parameters. If you are interested in conducting statistical inference on the resulting slope coefficients and require confidence intervals, then you should use the slower `fe1/x2` approach instead.

## Interactions that don't involve fixed-effects

The preceding interaction examples all took place in the fixed-effects slot (i.e. after the `|`). Underneath the hood, methods like `fe1^fe2` and `fe1[x2]` use numerical shortcuts that greatly speed up computation. This comes at the cost of being able to do richer inference with the interacted terms --- e.g. construct confidence intervals --- since they are swept out of the model at estimation time. In many cases, this should prove to be a perfectly acceptable trade-off. After all, the primary motivation for estimating fixed-effects models in the first place is to remove the confounding effect of such "nuisance" parameters. 

Yet, there are equally many cases where we do want to retain full inferential information about interaction terms. For example, in a difference-in-differences (DiD) research design. Here too `fixest` offers its own specialized syntax for combining terms in the model formula: `i()`. 

As we shall see, `i()` is more versatile than just creating interaction terms. It is also able to coerce individual variables to a factor and, compared to base R `factor()`, does so in a way that allows for better synergies with other `fixest` functions like `etable` and `coefplot`/`iplot`. This will allow us to perform some common estimation tasks much more easily.

We now illustrate the main uses of `i()` with the help of some examples.

### A compact syntax for factors and interactions: `i()`

The function `i()` accepts the arguments `ref`, and `keep` which are used to identify which levels to keep. Here's an example how to use them:

```{r}
data(airquality)
res_i1 = feols(Ozone ~ Solar.R + i(Month), airquality)
res_i2 = feols(Ozone ~ Solar.R + i(Month, ref = 8), airquality)
res_i3 = feols(Ozone ~ Solar.R + i(Month, keep = 5:6), airquality)

etable(res_i1, res_i2, res_i3, dict = c("6" = "June", "Month::5" = "May"), 
       order = c("Int|May", "Mon"))
```

Now that we've seen the basic `i()` functionality with a single variable, let's see how it can be put to use in an interaction context. We will consider two DiD examples of increasing complexity.

### Simple difference-in-differences (TWFE)

To illustrate `i()`'s interaction functionality, we will estimate treatment effects over time in a simple DiD setup. 

Consider the following dataset where some individuals (identified by the variable `id`) receive a treatment and others do not (as represented by the variable `treat`). The treatment takes effect from `period` 6 onward. The data look like this:

```{r, eval = TRUE}
# Sample data illustrating the DiD
data(base_did)
head(base_did)
```

The empirical approach that we will use here is the canonical two-way fixed effects (TWFE) model that should be familiar to many empirical researchers. In a regression context, TWFE essentially amounts to an interaction between our `treat` and `period` variables. This is easily done using the `i(fact_var, num_var, reference)` syntax:

```{r, eval = TRUE}
# Estimation of treatment × period effects
# We also add individual and period fixed-effects:
est_did = feols(y ~ x1 + i(period, treat, 5) | id + period, base_did)
est_did
```

Here, `i(period, treat, 5)` creates a set of interact terms: One `treat` dummy for each `period`, but excluding period 5 as a reference. While the explicit reference period is optional, note that we need a reference to avoid the collinearity problem posed by the `id` fixed effects. More to the point, for this DiD analysis we are interested in whether the treatment generated a sustained impact on our outcome variable relative to the control group. This makes period 5 (immediately prior to treatment) the obvious choice to use as our relative anchor. It also demonstrates an advantage that `i()`'s syntax offers over the equivalent base `treat*factor(period)` expansion, which would automatically use period 1 as a reference. Of course, we could specify the levels of `factor(period)` ahead of time. But this is rather tedious, whereas `i()` allows us to do so concisely on the fly.

To display the treatment effects on a graph, we can use the function `iplot`:

```{r, fig.width=7}
iplot(est_did)
```

The function `iplot` reports the coefficients of variables created with `i()`, and only them. In case you wanted to display all the coefficients, including `x1`, you can use the function `coefplot` instead.

### Staggered difference-in-differences (Sun and Abraham, 2020)

*Note that the content of this section is only valid for `fixest` version 0.9.0 or later.*

Despite TWFE's long-standing popularity for DiD designs, recent research has a shone a light on the biases that it can introduce under differential timing. For example, under staggered treatment or policy rollout. 

For staggered DiD analyzes, `fixest` provides the `sunab` function for implementing the unbiased method of [Sun and Abraham (2020)](https://doi.org/10.1016/j.jeconom.2020.09.006) --- hereafter: SA20. The method is based on the estimation of `treated cohort x time to treatment` dummies which are later aggregated to obtain the average treatment for the treated (ATT) for each period (or for the full post treatment period).

(Aside: Note that we use `time to treatment` rather than just `time`, because we need to account for the staggered treatment via a relative time measure.)

Let's walk through an example with the `base_stagg` data set, where (a) different cohorts get the treatment at different periods, and (b) the effect of the treatment is cohort-specific and increasing over time. Let's have a first look at the data:

```{r}
data(base_stagg)

head(base_stagg)
```

Note that the variable `year_treated` is set to 1000 (i.e. above the maximum value of `year`) for the never treated. Here's a quick visual depiction of the data. Again, notice the staggered treatment rollout and time-varying intensity.

```{r, echo=FALSE}
if(requireNamespace("ggplot2", quietly = TRUE)){
  library(ggplot2)
  ggplot(aggregate(base_stagg[, c('year_treated', 'treatment_effect_true')], 
                   by = list(year = base_stagg$year, group = to_integer(base_stagg$year_treated)), 
                   mean), 
         aes(year, group, fill = year>=year_treated, alpha = treatment_effect_true)) + 
    geom_tile(colour = "white", lwd = 1) + 
    scale_fill_brewer('Treated?', palette = 'Set1') +
    scale_alpha('Avg. treatment\neffect') +
    labs(x = 'Year', y = 'Group') +
    theme_minimal()
} else {
  print("This graph requires ggplot2 which is currently not available.")
}
```

With our data in hand, let's move on to the estimations. We'll implement two DiD models:

  1. A "naive" DiD consisting of a standard TWFE interaction (i.e. treatment × time_to_treatment).
  2. A DiD using the SA20 method.

```{r}
# "Naive" TWFE DiD (note that the time to treatment for the never treated is -1000)
# (by using ref = c(-1, -1000) we exclude the period just before the treatment and 
# the never treated)
res_twfe = feols(y ~ x1 + i(time_to_treatment, ref = c(-1, -1000)) | id + year, base_stagg)

# To implement the Sun and Abraham (2020) method,
# we use the sunab(cohort, period) function
res_sa20 = feols(y ~ x1 + sunab(year_treated, year) | id + year, base_stagg)
```

Let's compare our results in plot form:

```{r}
# Plot the two TWFE results
iplot(list(res_twfe, res_sa20), sep = 0.5)

# Add the true results
att_true = tapply(base_stagg$treatment_effect_true, base_stagg$time_to_treatment, mean)[-1]
points(-9:8, att_true, pch = 15, col = 4)

legend("topleft", col = c(1, 4, 2), pch = c(20, 17, 15), 
       legend = c("TWFE", "Truth", "Sun & Abraham (2020)"))
```

As we can see, the TWFE results suffer from varying degrees of bias due to the staggered rollout. In contrast, the aggregated SA20 results do a much better job of consistently estimating the true effects.

As a final example, here's how to get the average treatment effect for the treated (ATT):

```{r}
# The full ATT
summary(res_sa20, agg = "att")
```

This time the aggregation of the coefficients is operated across each cohort for non-negative time to treatment. Note that the estimated ATT compares favourably with the true value of `r with(base_stagg, mean(treatment_effect_true[time_to_treatment >= 0]))`.

Under the hood, the `sunab()` function creates `cohort x time to treatment` interactions with the function `i()`, the coefficients are then aggregated at the appropriate level with the function `aggregate.fixest`. To obtain the coefficients for each `cohort x time to treatment`, one simply needs to remove the aggregation:

```{r}
# Full disaggregation (you could have used summary instead of etable)
head(etable(res_sa20, agg = FALSE), 20)
```


# Additional features 

Now we present some other features of the package:

  1. [Formula: macros and dot square brackets](#formula-macros). 
  
  2. [How to add lead/lags](#lagging-variables).
  
  3. [Non-linear in parameter estimation](#non-linear-in-parameters-example). 
  
  4. [Parallelism](#multi-threading). 
  
  
<!-- NOTE: you can't have ":" or "(" in titles to make links -->


## Formula: macros and dot square brackets

### Formula macros

It can be useful, to tidy up the workflow or simply for convenience, to define formula macros: i.e. macro variables each representing several variables. This package offers two functions to deal with that: `setFixest_fml` to set macro variables globally, and `xpd` to create a formula with macro variables either globally defined or defined at run time.

Here is an example:
```{r}
base = iris
names(base) = c("y", "x1", "x2", "x3", "species")
# Defining the macro variables
setFixest_fml(..ctrl = ~poly(x2, 2) + poly(x3, 2))
# Accessing them
xpd(y ~ x1 + ..ctrl)

# Definition at run time
vars = c("x2", "x2^2", "x3")
for(i in 1:3){
  print(xpd(y ~ x1 + ..ctrl, ..ctrl = vars[1:i]))
}
```

Macro variables must start with two dots. As seen in the example, once set globally they can be directly accessed with `xpd`. In the example, `xpd` redefines the formula at run-time by having a character vector as input. 

All `fixest` estimations call `xpd` internally if appropriate. This means that the following code works just fine:
```{r}
feols(y ~ x1 + ..ctrl, base)
```

The function `xpd` also accepts regular expression macros of the form `..("regex")`. If these macros are encountered and provided a character vector/data.frame is in the argument `data`,  then all variables names matching the regular expression will be placed in the formula.

```{r}
data(longley)
xpd(Armed.Forces ~ Population + ..("GNP|ployed"), data = longley)
```

Again, since `xpd` is called internally, the following works:

```{r}
feols(Armed.Forces ~ Population + ..("GNP|ployed"), longley)
```

### The dot square bracket operator

The special operator `.[]` can be used in `fixest` formulas to replace the enclosed item by its value in the current environment. Let's give an example:
```{r}
base = setNames(iris, c("y", "x1", "x2", "x3", "species"))
i = 2:3
z = "i(species)"
feols(y ~ x.[i] + .[z], base)
```
What did just happen? When `.[]` is encountered in a formula, its value is evaluated and inserted verbatim in the formula. Further, when the value in brackets is a vector, as many variables are added in the model. That's why `x.[i]` led to `x2 + x3`. Of course, numbers can be used directly: `x.[2:3]` would have led to the same result.

Variable names in which `.[]` is in the middle of the name can also be created, but the use of macros is then mandatory:
```{r}
i = 1:5
xpd(y ~ ..x, ..x = "x.[i]_sq")
```
The use of a character string is mandatory here to avoid formula parsing problems.


## Lagging variables

Some methods have been created to easily deal with leads and lags in a panel setting. First is shown the various ways of how to deal with it in a `fixest` estimation, then how to create a panel data set, last is described a general way to create lead and lag variables.

### Lagging variables in an estimation

There are two ways to provide leads and/or lags in a `fixest` estimation. Either "on-the-fly" by using the argument `panel.id`, either by setting up a panel data set with the function `panel` (useful if you run multiple models needing lags and/or leads), described in the next subsection.

To set up the panel on-the-fly, you need to use the `panel.id` argument, which will give you access to the functions `f` and `l` to create leads and lags, respectively. Let's give an example, using the previous data set, where we use lags and leads of the explanatory and dependent variables.

```{r}
est1 = feols(y ~ l(x1, 0:1), base_did, panel.id = ~id+period)
est2 = feols(f(y) ~ l(x1, -1:1), base_did, panel.id = ~id+period)
est3 = feols(l(y) ~ l(x1, 0:3), base_did, panel.id = ~id+period)
etable(est1, est2, est3, order = "f", drop = "Int")
```

First of all, the value of `panel.id` corresponds to panel identifiers: first comes the variable identifying the panel-individuals, then comes the variables identifying the time. In the example, the `panel.id` came in the form of a one-sided formula, but it could equivalently be equal to `panel.id = c("id", "period")` or `panel.id = "id,period"`.

As we can see, in the first estimation, a lag is created with the function `l`. The second argument of the function consists of a vector giving the lags we want. In this case, `l(x1, 0:1)` means we want `x1` and the first lag of `x1`. This argument accepts negative values which give leads, as shown in the second estimation where `l(x1, -1:1)` means that we want the first lead of `x1`, `x1`, then its first lag. In this example, we also use the first lead of `y` as a dependent variable with `f(y)`.

### Setting up a panel data set

You can set up a panel data set with the function `panel`. There are between two and three advantages of doing so. First, if you have to estimate many models with leads and/or lags, setting up a panel gives you access to the lagging functions, `f` and `l`, without having to provide the `panel.id` argument we've seen in the previous subsection. Second, even when you estimate models with a sub-selection of the panel data, the data set is still a proper panel, meaning you can still use the lagging functions. Finally, and only if you are a `data.table` user, you can use the lagging functions to create new variables.

Let's repeat the previous example, now with a panel data set:
```{r}
# setting up the panel
pdat = panel(base_did, ~id + period)
# Now the panel.id argument is not required
est1 = feols(y ~ l(x1, 0:1), pdat)
est2 = feols(f(y) ~ l(x1, -1:1), pdat)
# You can use sub selections of the panel data
est_sub = feols(y ~ l(x1, 0:1), pdat[!pdat$period %in% c(2, 4)])
etable(est1, est2, est_sub, order = "f", drop = "Int")
```

Last, if you also use `data.table`, let's give an example of lead/lag creation:

```{r, eval = is_DT}
library(data.table)
pdat_dt = panel(as.data.table(base_did), ~id+period)
# we create a lagged value of the variable x1
pdat_dt[, x1_l1 := l(x1)]
# Now 
pdat_dt[, c("x1_l1_fill0", "y_f2") := .(l(x1, fill = 0), f(y, 2))]
head(pdat_dt)
```

### Creating lag/lead variables

If you just want to create a single lag/lead variable, a simple and fast method has been implemented: `lag.formula`. Let's give an example:

```{r}
base_lag = base_did
# we create a lagged value of the variable x1
base_lag$x1.l1 = lag(x1 ~ id + period, 1, base_lag)
head(base_lag)
```

The first two arguments are mandatory. The formula informs on the variable to be lagged (on the left hand side), and the two panel identifiers. Note that the time index **must** appear second. The second argument tells how much lags we want. Using negative values gives leads. Finally the last argument informs on where to find the variables.

In case you use the popular package `data.table`, you can create lagged variables very simply:

```{r, eval = is_DT}
library(data.table)
base_lag_dt = as.data.table(base_did)
# we create a lagged value of the variable x1
base_lag_dt[, x1.l1 := lag(x1 ~ id + period, 1)]
```

## Non-linear in parameters example

The function `feNmlm` is similar to `femlm` but allows to have non-linear in parameters right-hand-sides (RHS). First an example without fixed-effects, the one with fixed-effects is given later. 
Let's say we want to estimate the following relation with a Poisson model:

$E\left(z_i\right) = a\times x_i + b\times y_i$.

In fact, this type of model is non-linear in the context of a Poisson model because the sum is embedded within the log:

$E\left(z_i\right) = \exp\left(\log\left(a\times x_i + b\times y_i\right)\right)$.

So let's estimate such a relation. (Note that you can estimate this relation with GLM and identity link, but I carry on for the example.) First we generate the data:

```{r}
# Generating data:
n = 1000
# x and y: two positive random variables
x = rnorm(n, 1, 5)**2
y = rnorm(n, -1, 5)**2
# E(z) = 2*x + 3*y and some noise
z = rpois(n, 2*x + 3*y) + rpois(n, 1)
base = data.frame(x, y, z)
```

To estimate the non-linear relationship, we need to use the argument `NL.fml` where we put the non-linear part. We also have to provide starting values with the argument `NL.start`. Finally, to ensure the RHS can be evaluated in any situation, we add lower bounds for the parameters with the argument `lower`.

```{r}
result_NL = feNmlm(z~0, base, NL.fml = ~ log(a*x + b*y), NL.start = list(a=1, b=1), lower = list(a=0, b=0))
```

Note that the arguments `NL.start` and `lower` are named lists. Setting `lower = list(a=0, b=0)` means that the optimization algorithm will never explore parameters for $a$ and $b$ that are lower than 0.
The results obtained can be interpreted similarly to results with linear RHS. We can see them with a print:

```{r}
print(result_NL)
```

We can see that we obtain coefficients close to the generating values.

### Adding fixed-effects to non-linear in parameters models

Adding fixed-effects is identical to the linear case. The user must only be well aware of the functional form. Indeed, the fixed-effects must enter the estimation **linearly**. This means that the previous equation with one set of fixed-effects writes:

$E\left(z_i\right) = \gamma_{id_i} \left( a\times x_i + b\times y_i \right)$,

where $id_i$ is the class of observation $i$ and $\gamma$ is the vector of fixed-effects. Here the fixed-effects are in fact linear because in the context of the Poisson model we estimate:

$E\left(z_i\right) = \exp\left(\gamma_{id_i}+\log\left(a\times x_i + b\times y_i\right)\right)$.

Further, remark that there exists an infinity of values of $\gamma^{\prime}$, $a^{\prime}$ and $b^{\prime}$ such that:

$\gamma_{k} \left( a\times x_i + b\times y_i \right) = \gamma_{k}^{\prime} \left(a^{\prime}\times x_i + b^{\prime}\times y_i \right),\forall i,k$.

An example is $\gamma^{\prime}_{k} = 2\times \gamma_k$, $a^{\prime} = a/2$ and $b^{\prime} = b/2$. Thus estimating this relation directly will lead to a problem to uniquely identify the coefficients. To circumvent this problem, we just have to fix one of the coefficient, this will ensure that we uniquely identify them. 

Let's generate this relation:

```{r}
# the class of each observation
id = sample(20, n, replace = TRUE)
base$id = id
# the vector of fixed-effects
gamma = rnorm(20)**2
# the new vector z_bis
z_bis = rpois(n, gamma[id] * (2*x + 3*y)) + rpois(n, 1)
base$z_bis = z_bis
```

Now we estimate it with the fixed-effects while fixing one of the coefficients (we fix $a$ to its true value but it could be any value):

```{r}
# we add the fixed-effect in the formula
result_NL_fe = feNmlm(z_bis~0|id, base, NL.fml = ~ log(2*x + b*y), NL.start = list(b=1), lower = list(b=0))
# The coef should be around 3
coef(result_NL_fe)
# the gamma and the exponential of the fixed-effects should be similar
rbind(gamma, exp(fixef(result_NL_fe)$id[as.character(1:20)]))

```

As we can see, we obtain the "right" estimates. 


## Multi-threading

The package `fixest` integrates multi-platform parallelism to hasten the estimation process. By default it makes use of 50% of all available threads. To change the number of threads used, just use the argument `nthreads`:

```{r, eval = FALSE}
# Sample of results:
# 1 nthreads: 3.13s
system.time(fenegbin(Euros ~ log(dist_km)|Origin+Destination+Product+Year, trade, nthreads = 1))
# 2 nthreads: 1.82s
system.time(fenegbin(Euros ~ log(dist_km)|Origin+Destination+Product+Year, trade, nthreads = 2))
# 4 nthreads: 1.17s
system.time(fenegbin(Euros ~ log(dist_km)|Origin+Destination+Product+Year, trade, nthreads = 4))
```

As you can see, the efficiency of increasing the number of threads is not 1 to 1. Two threads do not divide the computing time by 2, nor four threads by 4. However it still reduces significantly the computing time, which might be valuable for large sample estimations.

You can permanently set the number of threads used by `fixest` using `setFixest_nthreads(nthreads)`. Note that you can permanently set this number at the project level by using the argument save: `setFixest_nthreads(nthreads, save = TRUE)`. This way you won't need to re-set the number of threads whenever you restart *R*.





